Question

If random samples of the given sizes are drawn from populations with the given proportions: (a) Find the standard error of the distribution of differences in sample proportions, \(\hat{p}_{A}-\hat{p}_{B}\) (b) Determine whether the sample sizes are large enough for the Central Limit Theorem to apply. Samples of size 50 from population \(A\) with proportion 0.70 and samples of size 75 from population \(B\) with proportion 0.60

Short answer

The standard error of the distribution of differences in sample proportions, \(\hat{p}_{A}-\hat{p}_{B}\) can be found using the given formula, and the Central Limit Theorem can be applied if both \(n*p>5\) and \(n*(1-p)>5\) for both the samples.

Step-by-step solution

Determine the Variables

Identify the variables from the problem:\(n_A = 50\), \(p_A = 0.70\), \(n_B = 75\), \(p_B = 0.60\)

Calculate the Standard Error

The standard error for the distribution of differences in sample proportions, \(SE(\hat{p}_A - \hat{p}_B)\) is given by the formula: \[SE(\hat{p}_A - \hat{p}_B) = \sqrt{{p_A(1 - p_A)/n_A + p_B (1 - p_B) / n_B}}\]Substituting the given values into the formula, \[SE = \sqrt{{0.70(1 - 0.70)/50 + 0.60(1 - 0.60) / 75}}\]

Evaluate the Expression

Evaluate the expression inside the square root to obtain the standard error. This would require multiplying the terms inside the brackets, dividing by the respective sample sizes and adding the results.

Apply Central Limit Theorem Conditions

The Central Limit Theorem can be applied if the sample size is large enough, which generally means \(n*p>5\) and \(n*(1-p)>5\). Check these inequalities for both sample A and B.

Key concepts

Standard Error of Sample Proportions

Understanding the standard error of sample proportions is key when dealing with statistics that involve comparing groups. This measure gives us insight into the variability one can expect when comparing sample proportions from two different populations. In the given exercise, to calculate the standard error of the difference in sample proportions, \( \hat{p}_{A} - \hat{p}_{B} \), we use the formula:
\[SE(\hat{p}_A - \hat{p}_B) = \sqrt{{p_A(1 - p_A)/n_A + p_B (1 - p_B) / n_B}}\]
Here, \( p_A \) and \( p_B \) are the sample proportions, and \( n_A \) and \( n_B \) are the sample sizes from the two different populations A and B, respectively. Once values are substituted and the mathematical operations are carried out, one obtains the standard error, which quantifies the average amount that the difference in sample proportions will deviate from the true difference in population proportions due to random sampling.

To put it simply, a smaller standard error indicates that the sample proportions are more likely to closely represent the true population proportions.

Distribution of Differences in Sample Proportions

When we compare sample proportions, like in this exercise, we're not just computing a single figure, but actually referring to a whole distribution of differences. This distribution tells us how the sample proportions from two populations could vary from one another by mere chance. The Central Limit Theorem (CLT) assures us that, provided certain conditions are met, this distribution will approximate the normal distribution, regardless of the underlying population's distribution.
In practical terms, having a normally distributed set of differences allows us to make inferences about the population using the familiar properties of the normal curve, such as confidence intervals and hypothesis testing. The calculation of the standard error, which we previously discussed, is a fundamental step in evaluating this distribution and hence in making reliable conclusions about the populations from our samples.

Sample Size Criteria for CLT

Applying the Central Limit Theorem (CLT) hinges on having a large enough sample size. This 'large enough' condition is typically quantified by a set of criteria that the sample must satisfy. Generally, these criteria are that each possible outcome of an event must occur more than a certain number of times on average for the theorem to apply. A commonly used rule of thumb is that both \( n*p \) and \( n*(1-p) \) must be greater than 5.
For our given problem, this means checking if \( n_A*p_A \) and \( n_A*(1-p_A) \) for population A, and similarly \( n_B*p_B \) and \( n_B*(1-p_B) \) for population B, are greater than 5. If these conditions are met for both samples, the CLT can be used, and we can proceed with confidence knowing the sampling distribution of the sample proportions will be approximately normal. This approximation allows for the practical application of statistical tests and intervals to make inferences about the population parameters from our sample statistics.